I was wondering what exactly the term Fourier symbol, or sometimes said the symbol of the Fourier transform means. Also, are any books that cover this topic in detail?
Edit:After the comments, I have come to realise I mean to ask either: what is the symbol of the Fourier transform operator, or also known as Fourier multiplier.
Are there are good recommended texts that cover the topic of symbols of operators, and the Fourier transform, well?
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$\begingroup$Probably you are interested in the symbol $P$ of of a pseudo-differential operator $T$ on $\mathbb R^n$ (or other...), the latter described by (up to choices)$$ Tf(y) \;=\; \int_{\mathbb R^n} e^{i\langle y,\xi\rangle}\; P(\xi,y) \Big(\int_{\mathbb R^n} e^{-i\langle \xi,x\rangle} \; f(x)\;dx\Big)\;d\xi $$This generalizes (in one way) the idea that Fourier transform converts constant-coefficient differential operators to multiplications by polynomials $P(\xi)$.
G. Folland's books (both Tata intro to PDEs and fuller textbook on PDEs) give good settings for and intros to simple cases of pseudo-differential operators. Many other sources (such as L. Hormander's books on PDE) treat general cases, which may obscure points of interest for a person new to the ideas.
$\endgroup$ 1 $\begingroup$My guess is it's the $\mathcal{F}$ or $\tilde{}$ in definitions such as$$\mathcal{F}(f(x))(k):=\tilde{f}(k):=\int_{\Bbb R^n}f(x)\exp(-2\pi ik\cdot x)d^nx.$$
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